Gaussian Quadrature

The Newton-Cotes formulae were of the form

where the were equidistant points in the integration interval

. The constants could be constructed such that the formula would be exact for solving polynomials of a degree smaller or equal to . In fact, formula's with an odd number of points achieve a degree of precision equal to .

Can one do better by choosing the points as well? Obviously, this would lead to a formula which can only be used if the function was known.

If both and can be chosen then we have unknown parameters in the formula. This is the same as the number of coefficients in a polynomial of degree . Could it be possible that one can determine the and such that the formula would be exact for polynomials up to degree ?

To answer this question, we turn to the theory of vector spaces of functions.

Preliminaries

Theorem 17 (Gaussian Quadrature)

Application

Examples

Integration over (a,b)